Remark 0.2 The definition of \(\nabla ^{m}A\)  is very subtle. In ([1]},[2]}), Huisken's definitions about \(\nabla ^{m} A\)  only work in the hypersurface's case. Andrews-Bakers ([3]}) gave a rigorous definition of \(\nabla ^{m} A\)  for any mean curvature flow with arbitrary codimension in Riemannian manifold. The basic idea is that \(|\nabla ^{m}A|\)  can contain all information about the mean curvature flow. Namely, when \(|\nabla ^{m}A|\)  goes to 0 when \(t\)  goes to infinity, the mean curvature flow \(F_{t}\)  will converge to a geodesic submanifold in some natural sense. For example, one of these cases is that \(F_{t}\)  has long time existence and always stay in a compact region of its ambient Riemannian manifold. We will state the definition of \(\nabla ^{m}A\)  in Section 2.

